## 1. Introduction

[2] In recent years, data assimilation techniques have been used to increase one's ability to predict and characterize atmospheric chemical phenomena by providing valuable estimates of surface emissions, improved model sensitivities, and optimized measurement strategies. By enforcing closure between model predictions and experimental observations, these methods constrain the variance of chemical transport models (CTMs) to produce optimal representations of the state of the atmosphere. As the number of variables used to describe the state of the atmosphere increases, the process of integrating models and measurements becomes increasingly difficult. Fortunately, advances in algorithm efficiency, computational resources, and the theory of inverse modeling have facilitated extension of these techniques to systems of increasing complexity. Anticipating the point at which all main features of sophisticated atmospheric CTMs are endowed with an inverse, this work examines the possibilities of extending data assimilation studies to include explicit consideration of size and composition aerosol dynamics.

[3] Although the actual implementation of data assimilation methods can be quite different, in general all techniques utilize some observational data set to provide an improved model representation of the system in question. Many previous studies on inverse modeling have utilized the Kalman filter, wherein propagation of the error covariance matrix is used to retain consistency between the model and the measurements [*Lyster et al.*, 1997; *Khattatov et al.*, 2000; *Stajner et al.*, 2001; *Palmer et al.*, 2003a]. While using a Kalman filter has the distinct advantage that model error is explicitly included in the analysis, the large computational cost of this approach has historically been the prime motivation for development of other methods. As an alternative approach, the adjoint method was first suggested as an efficient technique for performing variational data assimilations in atmospheric transport models by *Marchuk* [1974]. Originating from the mathematics of systems optimization and control theory [*Cacuci*, 1981a, 1981b] and well established in the fields of fluid mechanics [*Pironneau*, 1974], meteorology [*Talagrand and Courtier*, 1987], and oceanography [*Tziperman and Thacker*, 1989], the adjoint method has only been applied to CTMs relatively recently [*Fisher and Lary*, 1995; *Elbern et al.*, 1997; *Errera and Fonteyn*, 2001]. The treatment, while successful, has been limited to the assimilation and recovery of gas-phase species.

[4] The inclusion of detailed aerosol chemistry and physics has become requisite in atmospheric CTMs. Future implementation of four-dimensional variational analysis (4D-Var) assimilation techniques will likewise require the inclusion of aerosols in the adjoint models. To lay the groundwork for this endeavor, the fundamental capabilities (and limitations) of applying such techniques to aerosols need to be investigated. In this paper, we apply the first inverse models of multicomponent aerosol dynamics and evaluate their performance under conditions designed to facilitate incorporation of these routines into existing adjoint CTMs. A paper presenting derivations of the necessary equations for several other forms of inverse aerosol models and evaluation of these for a simple, single-component aerosol has also been submitted (A. Sandu et al., Inverse modeling of aerosol dynamics using adjoints: Theoretical and numerical considerations, submitted to *Mathematics and Computers in Simulation*, 2004) (hereinafter referred to as Sandu et al., submitted manuscript, 2004). These works differ substantially from the only previous data assimilation study involving aerosols [*Collins et al.*, 2001] in that the aerosol distribution is allowed to evolve according to the aerosol dynamic equation [*Pilinis*, 1990] and that the inversion is performed using the adjoint technique. In the study by Collins et al. the aerosols were represented as growing via empirical correlations and growth rates, and the total aerosol optical depth was assimilated sequentially using a Kalman filter.

[5] With the above goal in mind, adjoint aerosol models are developed and are tested using simulated observations (commonly known as an identical twin experiment). The (forward) aerosol model used is a simplified, yet numerically and physically consistent, version of the aerosol submodel currently employed in several four-dimensional (4-D) CTMs [*Meng et al.*, 1998; *Song and Carmichael*, 2001]. As operator splitting is used in such models to isolate all aerosol processes into a single 0-D (box) routine, which is called within each cell of the discretized 3-D spatial field, it is sufficient to use a forward box model that does not include gas-phase chemistry or spatial advection. Within this forward box model, emphasis is placed on gas-to-particle conversion, wherein gas-phase transport is the rate-limiting step for particle growth. The details of the forward model are given in section 2.

[6] An immediate application of an inverse aerosol model is to infer the size distributions of aerosol sources using surface, airborne, or possibly even satellite measurements. This involves reconstructing back trajectories of the distribution by repetitive calls to the adjoint box model from within the overall adjoint 4-D CTM, asking each time to recover the shape of the distribution at a previous time step. Therefore an important capability of the aerosol adjoint routine is to recover an initial size distribution on the basis of knowledge of the distribution at some later time(s). The length of the assimilation period will depend upon the temporal resolution of the forward model and the frequency of the observations; herein we consider periods ranging from several minutes to a few hours.

[7] In addition to recovering initial distributions, an inverse aerosol model can be used to estimate physical properties key to the dynamic evolution of the distribution by treating these quantities as variable parameters. The growth of aerosol particles due to condensation/evaporation is heavily influenced by the thermodynamic properties of the transferring species. A significant fraction of organic aerosol particles is composed of chemical compounds whose thermodynamic properties in the particulate phase are not well characterized. Better estimates of such properties would not only increase the accuracy of CTMs but also aid in interpretation of laboratory studies of aerosol dynamics. Hence another desired capability of an adjoint aerosol model is to provide estimates of the thermodynamic properties of the aerosol species.

[8] The aerosol adjoint models can also help refine experimental measurement strategies. Conditions can be simulated in which either individual species are not measured or the size distribution is only partially sampled. Comparison of the assimilations between these scenarios leads to sampling schemes that provide an optimum balance between data recoverability and observational burden.

[9] One of the primary reasons for choosing the adjoint method to construct an inverse aerosol model is the computational efficiency of this approach. As variations in the actual implementation of this methodology affect the overall computational requirements, it is beneficial to consider different approaches to constructing the adjoint models, of which there are two generally recognized types: continuous and discrete [*Giles and Pierce*, 2000; *Tziperman and Thacker*, 1989]. The first method is to derive the continuous adjoint equations from the governing equations and then solve these numerically. The second approach is to cast the forward equations into a numerical discretized form and then take the adjoint of this discretized formula. Numerical discretization and adjoint operations do not commute in general; therefore the continuous and discrete approaches lead to final gradients that differ in accuracy and computational expense, and hence it is desirable to assess both tactics when introducing the adjoint method to a new field (Sandu et al., submitted manuscript, 2004).